Discrete Maths Help ASAP

Question

In a standard deck of cards, there are 4 suits: clubs, diamonds, hearts, and spades. Each suit contains 13 cards: 2, 3,…,10, Jack, Queen, King, Ace. So there are 4 * 13=52 cards in the deck.

A sequence of 5 cards is drawn from a standard deck of cards, with replacement (meaning that when a card is drawn it is placed back into the deck before the next card is chosen).

1) How many of such sequences begin with a King and end with a Queen?

2) How many of such sequences begin with a King or end with a Queen?

3) How many of such sequences will contain at least one King or one Queen (or both)?

Answer 1

1) Any one of the four kings can be first and any one of the queens second. In between, you can have any 3 of the 52 possible cards if you using with replacement. Therefore the number of ways equals ₄C₁*₅₂C₃*₄C₁

_{2) To begin with a king it is 4}C₁*₅₂C_{4; to end in a queen it is 52}C₄*₄C_1. Therefore to be either is ₄C₁*₅₂C₄ + ₅₂C₄*₄C_{1 - 4}C₁*₅₂C₃*₄C_1. You subtract to not double count if you begin with king and end in queen.

3) This is the compliment of not picking a queen or king, ₅₂C₅ - ₄₄C₅